Calculus II · Unit 4B · lesson

Radius and Interval of Convergence

Concept

Learning objectives

find the radius of convergence and describe the automatic convergence and divergence regions.

Radius and Interval of Convergence

Explanation

Power-series convergence has a rigid geometry

A power series has a remarkable all-or-nothing structure around its center. If it converges at a point some distance from the center, it converges absolutely at every closer point. If it diverges at one point, it diverges at every farther point. The result is a radius $R$ separating automatic convergence from automatic divergence.

The radius does not decide endpoint inclusion. At $|x-a|=R$ , the ratio or root calculation usually becomes exactly one and loses power. Each endpoint must be substituted into the original series and tested independently. The final interval may be open, closed, or half-open.

Bridge

Convergence spreads inward from the center

Power-series convergence is organized by distance from the center $a$ . If the series converges at a point $x_0\ne a$ , then it converges absolutely at every point closer to $a$ . If it diverges at one point, it diverges at every point farther away. This produces a radius $R$ and an interval centered at $a$ .

The radius settles only the interior and exterior. At the boundary points $a-R$ and $a+R$ , the geometric domination used inside has ratio one and stops deciding the question. Each endpoint must be tested separately as an ordinary numerical series.

**The radius organizes convergence by distance from the center**
A number line centered at a shows an interior interval of absolute convergence, two separately marked endpoints, and exterior divergence regions.
The radius organizes convergence by distance from the center. A number line with an absolute-convergence interior, undecided endpoints, and divergence exterior.

Decision

Interval-of-convergence workflow

First find the center and radius by testing absolute convergence for a generic $x$ . Record the open interior interval. Then substitute the left endpoint into the original series and test it; repeat independently at the right endpoint. Only after both decisions should interval notation be written.

Proof idea

A closer point introduces a geometric factor

If $\sum c_n(x_0-a)^n$ converges, its terms are bounded. For $|x-a|<|x_0-a|$ , write

|c_n(x-a)^n|=|c_n(x_0-a)^n|\left|\frac{x-a}{x_0-a}\right|^n.

The bounded first factor times a geometric factor with ratio below one proves absolute convergence.

**A power series converges on an interval centered at its expansion point**
Inside the radius $R$ , convergence is automatic; outside, divergence is automatic; each endpoint must be checked separately.
A power series converges on an interval centered at its expansion point. A number line with an absolute-convergence interior, undecided endpoints, and divergence exterior.

How to read the visual

The open left endpoint and closed right endpoint are only an example. Endpoint inclusion depends on separate one-variable series tests.

Concept

Radius structure

For some $R\in[0,\infty]$ , a power series centered at $a$ converges absolutely when $|x-a|<R$ and diverges when $|x-a|>R$ .

Guided walkthrough

A basic radius

For

\sum_{n=0}^{\infty}\frac{(x-2)^n}{3^n},

the geometric ratio is $(x-2)/3$ . Thus $|x-2|<3$ , so the radius is $R=3$ and the open convergence interval is $(-1,5)$ . Endpoints still require tests.

Worked example

Read the interval from the center and radius

Suppose a power series centered at $3$ has radius $5$ . It converges absolutely whenever

|x-3|<5,

which is the open interval $(-2,8)$ . It diverges for $x<-2$ or $x>8$ . The points $x=-2$ and $x=8$ remain unresolved until they are substituted into the original series.

Common mistake

Radius and interval are not the same object

The radius is the nonnegative number $R$ . The interval includes the center and records endpoint behavior, such as $[-2,8)$ .

Interactive checku4b-radius_and_interval_of_convergence-01

Find the radius of convergence of $\sum (x-2)^n/3^n$ .

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Show hint

Require the geometric ratio magnitude to be below one.

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Exercise

Find the radius of $\sum x^n/5^n$ .

Exercise

Find the center and radius of $\sum n(x+1)^n/4^n$ .

Exercise

Explain what $R=0$ and $R=\infty$ mean.

Exercise

Why does radius alone not settle endpoints?

After the explanation

Use the section idea

Reading lens

Find the radius from interior behavior, then test each boundary point as a separate numerical series.

Mental model

Distance from the center organizes the automatic interior and exterior behavior; endpoints remain independent decisions.

Decision

Use a ratio or root argument for the radius, convert it to an interval, and test both endpoints explicitly.

Common trap

Including or excluding both endpoints from the radius calculation alone.

Check yourself

Can you justify the radius and each endpoint with separate evidence?

Source & rights

Original instruction with traceable references.

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